I was wondering if there was a way to reinstall the 3.60 fw somehow (not the vita's internal "restore" but an actual reinstallation from the PUP or henkaku's DNS)

Thank you

Yes, many different ways. You can reboot into recovery and install the update via QCMA, gamecart (so long as it comes with 3.60), or the way I've been doing it while testing something on a logic board with a broken USB port, downloading the 3.63 firmware in the background, going in to ur0: and replacing the 3.63 pup with the 3.60 update pup. That way is pretty risky though so only do it if you have to.

To get it to download in the background make sure to turn on auto download updates, go into system update, back out and wait. Again, if you have a working usb port don't do this. Only as a last resort.